3.1 Robot design concept

The concept of robot design and motion planning mechanisms is inspired by the actions of wall-climbing athletes [38]. While athletes perform a climbing movement, the limbs that contribute include the joints of the arms, shoulders, and palms. Based on these considerations, we designed a transverse climbing robot with a four-link planar mechanism that offers three degrees of freedom (3 DOF) motion while holding onto the ledge with a single hand. In this study, the conceptual design model of the transverse climbing robot is shown in Fig. 3. The main components of this robot are two upper arms, two lower arms, and two grippers at each end of the upper arms. Both grippers are actuated by a motor servo. The other rotational joints are actuated by brushed DC motors to perform motion control.

Figure 3. Conceptual design model of a transverse ledge climbing robot; (a) isometric view (b) side view.

Previous generations of transverse ledge brachiation robots developed by Lin et al. [Reference Lin and Tian1, Reference Lin and Lee3, Reference Lin. and Yang39] performed movements by swinging the lower limb for energy accumulation. Different from this type of locomotion, our robot mechanism performs an exploratory movement by extending one of its hands towards the target position while the other hand grasps tightly to hold the robot on the ledge. Figure 4 illustrates a four-link robot moving through a gap transition between two ledges at different elevations. Compared to robots using a swing-like movement style, our robot does not need to go through the energy storage phase to initiate the whole motion and thus can directly extend the exploring arm for ledge grasping. However, while extending the arm, our robot requires higher joint torques and gripping forces to hold it on the ledge.

Figure 4. The locomotion of a transverse climbing robot.

The robot model used in this study was specifically designed for movement within a single plane aligned with the orientation and movement along a transverse ledge. We optimized the navigation to traverse horizontal ledges and climb within the plane of a high-rise building’s outer wall. Consequently, the robot is limited in its ability to adapt to changes in ledge orientation that are perpendicular to its plane of motion.

3.2. Robot dynamics

3.2.1. Lagrange-based dynamic model

The robot dynamic model is essential for joint motion control [Reference Olsen and Petersen40] and motion planning analysis. According to the Lagrange method [Reference Spong, Hutchinson and Vidyasagar41], a dynamic model of the transverse climbing robot could be written in the following standard form:

(1) \begin{align} M(q)\ddot{q}+C\left(q,\dot{q}\right)+G(q)+F_{f}(\dot{q})=\tau +\tau _{ext} \end{align}

Where $\mathit{q},\dot{\mathit{q}}$ , $\ddot{\mathit{q}}\in \mathit{R}^{\mathit{3}}$ represent the angular position, angular velocity, and angular acceleration, respectively; $\mathit{M}$ (q) $\ddot{\mathit{q}}\in \mathit{R}^{\mathit{3}\mathit{x}\mathit{3}}$ is the moment of inertia, $\mathit{C}$ ( $\mathit{q},\dot{\mathit{q}}$ ) $\in \mathit{R}^{\mathit{3}\mathit{x}\mathit{3}}$ is the centrifugal or Coriolis force; $\mathit{G}$ ( $\mathit{q}$ ) $\in \mathit{R}^{\mathit{3}\mathit{x}\mathit{1}}$ is the gravity force, $\mathit{F}_{\mathit{f}}$ ( $\dot{\mathit{q}}$ ) $\in \mathit{R}^{\mathit{3}\mathit{x}\mathit{1}}$ is the joint friction force, $\mathit{\tau }\in \mathit{R}^{\mathit{3}\mathit{x}\mathit{1}}$ is the joint torque; and $\mathit{\tau }_{\mathit{ext}} \in \mathit{R}^{\mathit{3}\mathit{x}\mathit{1}}$ is the external torque caused by external disturbances, especially the contact force between the robot end-effector and environment.

Figure 5 shows the schematic diagram of the four-link climbing robot, which has three degrees of freedom. Each link motion is joined by a revolving joint. The robot’s hands are each a returning hand in the grasp of a ledge (fixed joint) and an exploring hand in the release of its grasp (end-effector). To identify the robot’s kinematic link, we provide the Denavit–Hartenberg (DH) parameters in Table I [Reference Tsai42], where l is the link length, α is the twist link, d is the link offset, and q is the joint angle. The robot’s desired movement is in the right direction, according to a single plane. Table II summarizes the robot parameters’ symbols and approximated values obtained from the CAD software.

Table I. DH parameter of the transverse ledge climbing robot.

Figure 5. Schematic diagram of the transverse ledge climbing robot.

Table II. Robot model parameter.

3.2.2. Uncertain dynamic model

In practice, there are several changes in the parameters of the dynamic system when a robot performs the desired movements. This issue arises because of changes in the robot’s physical uncertainty, such as the center of gravity, moment of inertia, centrifugal or Coriolis, and joint friction [Reference Cao, Gan and Dai30]. Besides that, several other reasons may also cause dynamic uncertainty, such as noise in the sensor [Reference Long, Dang, Sun, Wang and Gui31], measurement inaccuracies, and a slight fault in calculating the dynamic system when using model-based control methods. The existence of dynamic uncertainty is the primary issue with robot control stability [Reference Song, Yi, Zhao and Li43, Reference Le, Kang, Suh and Ro44]. Even a slight consideration of this uncertainty is important to improving the stability and accuracy of robot movement. However, this uncertainty is difficult to predict and model, but it can still be modeled as an additional deviation from each parameter of the dynamic model, defined as follows [Reference Li, Li, Zhu, Xu and Mu28]:

(2) \begin{align} \left\{\begin{array}{l} M(q)\ddot{q}=M_{0}(q)\ddot{q}+\Delta M(q)\ddot{q}\\[3pt] C(q,\dot{q})=C_{0}(q,\dot{q})+\Delta C(q,\dot{q})\\[3pt] G(q)=G_{0}(q)+\Delta G(q)\\[3pt] F_{f}(\dot{q})=F_{f0}(\dot{q})+\Delta F_{f}(\dot{q}) \end{array}\right. \end{align}

where $\mathit{M}_{0}$ (q) $\ddot{\mathit{q}} ;\ \mathit{C}_{0}$ ( $\mathit{q}\mathit{,}\dot{\mathit{q}}$ ); $\mathit{G}_{0}$ (q) and $\mathit{F}_{\mathit{f}0}$ (q) are dynamic terms that can be modeled; while $\Delta \mathit{M}$ (q) $\ddot{\mathit{q}} ;\ \Delta \mathit{C}$ ( $\mathit{q}\mathit{,}\dot{\mathit{q}}$ ); $\Delta \mathit{G}$ (q) and $\Delta \mathit{F}_{\mathit{f}}$ (q) represent the unmodeled dynamics. The uncertain dynamic deviation $\rho$ can be written as follows:

(3) \begin{align} \rho =\Delta M(q)+\Delta C(q,\dot{q})+\Delta G(q)+\Delta F_{f}(\dot{q}) \end{align}

With the uncertain dynamic deviations, the robot model can be rewritten as follows:

(4) \begin{align} M_{0}(q)\ddot{q}+C_{0}(q,\dot{q})+G_{0}(q)+F_{f0}(\dot{q})=\tau +f\!\left(\tau _{ext},\rho, t\right) \end{align}

where $f(\tau _{ext},\rho, t)$ represent the lumped disturbance to the robot dynamic system.

3.3. Robot inverse kinematic

In a transverse climbing robot, the trajectory reference is based on the end-effector’s desired position. However, the robot’s motion is controlled by joint control. The inverse kinematics method [Reference Tsai42] is used to determine the angular position of each robot joint based on the position of the grasping and extended hands.

To facilitate the discussion, in Fig. 6, we present a simple illustrative diagram of the four-link transverse climbing robot for kinematic analysis. When the robot grasps the ledge, the grasping hand position causes the upper arm position to become perpendicular to the grasped ledge. Therefore, if there is a known inclination difference between the ledge grasped by each gripper, the robot’s geometry could be identified as follows:

Figure 6. Kinematic diagram of a four-link climbing robot with a slope environment.

(5) \begin{align} q_{1}+q_{2}+q_{3}=180^{\circ}+\beta \end{align}

Using the concept of forward kinematic analysis of the robot, the position of the gripper (Q) at the slope angle can be written as follows:

(6) \begin{align} \left\{\begin{array}{l} Q_{x}=l_{1}\cos\!q_{1}+l_{2}\cos\!\left(q_{1}+q_{2}\right)+l_{3}\cos\!\left(180^{\circ}+\beta \right)\,\\[3pt] Q_{y}=l_{1}\sin\!q_{1}+l_{2}\sin\!\left(q_{1}+q_{2}\right)+l_{3}\sin\!\left(180^{\circ}+\beta \right)\, \end{array}\right. \end{align}

The angular position at joint 2 is written as follows:

(7) \begin{align} q_{2}=\cos^{-1}\left(\frac{\left(Q_{x}+l_{3}\cos\! \left(\beta \right)\right)^{2}+\left(Q_{y}+l_{3}\sin\!\left(\beta \right)\right)^{2}-{l_{1}}^{2}-{l_{2}}^{2}}{2l_{1}l_{2}}\right) \end{align}

The angular position at joint 1 is written as follows:

(8) \begin{align} A=\frac{-\left(Q_{x3}\cos\!\beta \right)\left(l_{2}\sin\!q_{2}\right)+\left(Q_{y}+l_{3}\sin\!\beta \right)\left(l_{1}+l_{2}\cos\!q_{2}\right)}{{l_{1}}^{2}+{l_{2}}^{2}+2l_{1}l_{2}\cos\!q_{2}} \end{align}

(9) \begin{align} B=\frac{\left(Q_{x3}\cos\!\beta \right)\left(l_{1}+l_{2}\cos\!q_{2}\right)+\left(Q_{y}+l_{3}\sin\!\beta \right)\left(l_{2}\sin\!q_{2}\right)}{{l_{1}}^{2}+{l_{2}}^{2}+2l_{1}l_{2}\cos\!q_{2}} \end{align}

(10) \begin{align} q_{1}=\tan ^{-1}\left(\frac{A}{B}\right) \end{align}

If the angular position at joints 2 and 1 is known, the angular position at joint 3 can be obtained from Eq. (5).

4.1. First-order momentum observer (FOMO)

According to Leunberger’s observer definition [Reference Davis45], the observer consists of a state variable vector $\hat{\mathit{x}}$ and an output vector $\hat{\mathit{y}}$ . The state of the system to be estimated is momentum $\mathit{P}$ , where the equation can be expressed as follows:

(11) \begin{align} \,P(t)=M_{0}(q)\dot{q}(t) \end{align}

Then, the change in momentum per unit time $\,\dot{P}$ is defined as follow:

(12) \begin{align} \,\dot{P}=\dot{M}_{0}(q)\dot{q}(t)+M_{0}(q)\ddot{q}(t) \end{align}

Substitute the equations (4) and (12) as follows:

(13) \begin{align} \,\dot{P}(t)=\left(\dot{M}_{0}(q)-C_{0}(q,\dot{q})\right)\dot{q}(t)-\left(G_{0}(q)+F_{f0}(\dot{q})\right)+\left(\tau +f\left(\tau _{ext},\rho, t\right)\right) \end{align}

The matrix $(\dot{M}(q)-2C(q,\dot{q}))$ is skew-symmetric [Reference Johan, Gravdahl and Pettersen46], and the differentiation of the inertial matrix $\mathit{M}$ (q) could be written as:

(14) \begin{align} \dot{M}(q)=C\left(q,\dot{q}\right)+C^{T}\left(q,\dot{q}\right) \end{align}

Substitute equations (1) and (14) into Eq. (11), and the following is obtained:

(15) \begin{align} \dot{P}(t)\mathit{=}C^{T}\left(q,\dot{q}\right)\dot{q}(t)+\tau +\tau _{ext}-G(q)-\tau _{f} \end{align}

The equation of FOMO can be written as follows:

(16) \begin{align} \left\{\begin{array}{l} x_{1}(t)=P=M_{0}(q)\dot{q}(t)\\[3pt] \dot{x}_{1}(t)=\dot{P}=\tau _{p}^{*}(t)+f\left(\rho, \tau _{ext},t\right)=\left(\tau -G_{0}(q)-F_{f0}(\dot{q})\right)+\left(\tau _{ext}-\rho \right)\\[3pt] \hat{x}_{1}(t)=\hat{P}\\[3pt] \widehat{\dot{x}}_{1}(t)=\widehat{\dot{P}}=\tau +K_{I}e-G_{0}(q)-F_{f0}(\dot{q})\\[3pt] \hat{y}(t)=K_{I}\left(P-\hat{P}\right) \end{array}\right. \end{align}

where $\mathit{\tau }_{\mathit{p}}^{\mathit{*}}$ (t) is the internal disturbance force of the system, $\hat{y}(t)$ is the observer output as an estimator of the external disturbance force on the system, $\mathit{e}\mathit{=}$ ( $\mathit{P}-\dot{\mathit{P}}$ ) is the state estimation error, and $\mathit{K}_{\mathit{I}}$ is the observer gain constant for tuning the observer convergence. If the $\mathit{K}_{\mathit{I}}$ is set too high, it can amplify the noise, leading to increase in the estimation errors. Therefore, adjusting the observer gain requires consideration of observer’s bandwidth and noise level [Reference Garofalo, Mansfeld, Jankowski and Ott58].

4.2. 4.2. Second-order momentum observer (SOMO)

Because of the sensor limitations in the system, when a robot makes contact with its environment, the physical state cannot be measured directly, making it difficult to model accurately. Therefore, the extended state observer (ESO) is a solution to estimate the unmeasured physical state of a dynamic system [Reference Ren, Dong, Wu and Chen47]. We extend a state observer to SOMO by introducing additional state variables in FOMO. Making it could enhance the response’s performance in estimating the contact force. In this system, the equation for ESO variables is expressed as follows:

(17) \begin{align} \left\{\begin{array}{l} x_{1}(t)=P(t)=M_{0}(q)\dot{q}(t)\\[3pt] \dot{x}_{1}(t)=\tau *(t)+x_{2}(t)\\[3pt] \dot{x}_{2}(t)=\phi (t)\\[3pt] y(t)=x_{1}(t) \end{array}\right. \end{align}

Where $x_{2}$ is the new variable state of the SOMO, and $\phi$ is the unknown (unmeasured) disturbance. Thus, the SOMO equation can be written as follows:

(18) \begin{align} \left\{\begin{array}{l} \hat{x}_{1}(t)=\hat{P}(t)\\[3pt] \dot{\hat{x}}_{1}(t)=\tau (t)+\tau _{ext}(t)-\tau _{f0}\left(\ddot{q}\right)-G_{0}(q)+\hat{x}_{2}(t)+\beta _{1}e_{1}\\[3pt] \hat{x}_{2}(t)=\tau _{ext}(t)-\rho (t)=\hat{f}\left(\rho, \tau _{ext},t\right)\\[3pt] \dot{\hat{x}}_{2}(t)=\beta _{2}e_{1}\\[3pt] y(t)=\hat{x}_{2}(t)\\[3pt] e_{1}=\hat{x}_{1}-x_{1} \end{array}\right. \end{align}

Where $\mathit{e}_{1}\mathit{=} \hat{\mathit{x}}_{1}-\mathit{x}_{1}$ is the estimation error, $\mathit{\beta }_{1}$ and $\mathit{\beta }_{2}$ are the matrix observer gains for tuning the estimator convergence. The transfer function of the Laplace transforms from Eq. (18) can be written as follows:

(19) \begin{align} \hat{x}_{1}=\frac{\hat{x}_{2}+\beta _{1}p+\tau *}{\mathrm{s}+\beta _{1}} \end{align}

(20) \begin{align} \hat{x}_{2}=\frac{\beta _{2}p\left(\mathrm{s}+\beta _{1}-\beta _{1}\right)-\beta _{2}\tau _{p}^{*}}{\mathrm{s}^{2}+\beta _{1}s+\beta _{2}} \end{align}

where $\hat{\mathit{x}}_{1}$ and $\hat{\mathit{x}}_{2}$ represent lumped disturbance estimators. The Eq. (20) follows the characteristic of the second-order standard equation $\mathit{s}^{2} \mathit{+} 2\mathit{\xi }\mathit{\omega }_{\mathit{n}}\mathit{s}\mathit{+} {\mathit{\omega }_{\mathit{n}}}^{2}\mathit{=} 0$ . So, $\mathit{\beta }_{1}$ and $\mathit{\beta }_{2}$ can be obtained as follows:

(21) \begin{align} \left\{\begin{array}{l} \beta _{2}={\omega _{0}}^{2}\\[3pt] \beta _{1}=2\xi \omega _{0} \end{array}\right. \end{align}

where $\omega _{0}$ and $\xi$ represent the natural frequency and damping ratio of the system respectively, which determines the performance of the momentum observer.

4.3. Time-varying threshold (TVT)

The principle of TVT is to determine the tolerable force limit of an internal disturbance to avoid the possibility of misdetection [Reference Li, Li, Zhu, Xu and Mu28]. According to Eq. (3), the bounded model uncertainty can be expressed as a polynomial equation consisting of angular acceleration, angular velocity, angular position, and other disturbances that cannot be modeled.

(22) \begin{align} \left| \hat{\rho }_{i}\right| \leq \delta _{i}=b_{0}\left| \ddot{q}_{i}\right| +b_{1}\left| \dot{q}_{i}\right| +b_{2}\left| q_{i}\right| +b_{3}\mathrm{sgn}\left(\dot{q}_{i}\right) \end{align}

Where $\hat{\rho }$ is the model uncertainty that could be estimated while no external disturbance force is encountered by the robot, $\mathit{\delta }_{\mathit{i}}$ ( $\mathit{1}\leq \mathit{1}\leq \mathit{n}$ ) is a model uncertainty bound, and $\mathit{b}_{\mathit{0}},\mathit{b}_{\mathit{1}},\mathit{b}_{\mathit{2}},$ and $\mathit{b}_{\mathit{3}}$ is a polynomial coefficient. To determine the polynomial coefficient, the parameters of TVT can be identified by expressing the equation of TVT in the linear regression model as follows:

(23) \begin{align} \hat{\delta }=W\left(q_{d},\dot{q}_{d},\ddot{q}_{d}\right)\cdot \hat{\Theta } \end{align}

where $\,W$ is the regression matrix, while $\,\hat{\Theta }$ is the matrix coefficient. Each is defined as follows:

(24) \begin{align} W\left(q,\dot{q},\ddot{q}\right)=\left[\begin{array}{llll} \left| \ddot{q}_{di}\right| & \left| \dot{q}_{di}\right| & \left| q_{di}\right| & \mathrm{sgn}\left(\dot{q}_{di}\right) \end{array}\right] \end{align}

(25) \begin{align} \hat{\Theta }=\left[\begin{array}{l@{\quad}l@{\quad}l@{\quad}l} b_{0} & b_{1} & b_{2} & b_{3} \end{array}\right]^{T} \end{align}

When the robot moves along its trajectory, it must reidentify the $\hat{\Theta }$ to minimize the error in model uncertainty estimation by the parameter composition that consists of desired angular acceleration, desired angular velocity, desired angular position, and estimated model uncertainty. Thus, it can be written as follows:

(26) \begin{align} \hat{\Theta }=\left(W^{T}\cdot W\right)^{-1}W^{T}\cdot \hat{\delta } \end{align}

According to the TVT model, the upper threshold $\delta _{Ui}$ and lower threshold $\delta _{Li}$ can be expressed as follows:

(27) \begin{align} \left\{\begin{array}{l} \delta _{Ui}=W\left(q_{d},\dot{q}_{d},\ddot{q}_{d}\right)\cdot \hat{\Theta }+\lambda _{1}\\[3pt] \delta _{Li}=W\left(q_{d},\dot{q}_{d},\ddot{q}_{d}\right)\cdot \hat{\Theta }-\lambda _{2} \end{array}\right. \end{align}

where $\mathit{\lambda }_{1}$ and $\mathit{\lambda }_{2}$ are phase offset coefficients that can be adjusted depending on the intensity of model uncertainty estimation. The principle of the TVT is that when the value of the force disturbance estimation received by the robot exceeds the thresholds, the system will detect that force disturbance estimation comes from external sources or contact with the environment.

6.1. Computed torque control (CTC)

Computed torque control is able to compensate for some limited disturbance forces to be able to maintain robot stability [Reference Llama, Kelly and Santibanez48]. The fundament architecture of CTC involves an outer and an inner loop control system [Reference Shang and Cong49]. The outer loop system employs a proportional derivative to control the desired trajectory parameters, which include the desired joint position, desired velocity, and desired acceleration; and its feedback signal is the actual output trajectory. In the inner loop system, the real-time precise trajectory output is achieved by computing it based on the actual joint torque information (including disturbance torque) and the robot dynamic parameters.

According to Eq. (1), the dynamic model equation without lumped disturbance can be expressed as follows:

(28) \begin{align} M(q)\ddot{q}+C\left(q,\dot{q}\right)+G(q)=\tau \end{align}

The robot error dynamic can be expressed as follows:

(29) \begin{align} \left(\ddot{q}_{d}-\ddot{q}\right)+K_{d}\left(\dot{q}-\dot{q}_{d}\right)+K_{p}\left(q-q_{d}\right)=0 \end{align}

Then Eq. (29) can be substituted into Eq. (28) to get the joint torque input $\tau _{u}$ :

(30) \begin{align} M(q)\left[\ddot{q}_{d}-K_{d}\left(\dot{q}-\dot{q}_{d}\right)-K_{p}\left(q-q_{d}\right)\right]+C\left(q,\dot{q}\right)+G(q)=\tau _{u} \end{align}

If $e=q-q_{d}$ can be defined as the system error, Eq. (30) can be written as:

(31) \begin{align} \ddot{e}+K_{d}\dot{e}+K_{p}e=0 \end{align}

To stabilize the system, the proportional gain $\mathit{K}_{\mathit{p}}$ and derivative gain $\mathit{K}_{\mathit{d}}$ need to be determined. The results show that the system output can converge following the reference input. The characteristic Eq. (31) can be written as follows:

(32) \begin{align} \left| sI-A\right| =s^{2}+K_{d}s+K_{p}=0 \end{align}

The characteristics of Eq. (32) are similar to those of the standard second-order system. So, $\mathit{K}_{\mathit{p}}$ and $\mathit{K}_{\mathit{d}}$ can be written as follows:

(33) \begin{align} \left\{\begin{array}{l} K_{p}={\omega _{n}}^{2}\\[3pt] K_{d}=2\xi \omega _{n}\, \end{array}\right. \end{align}

where $\xi$ is the damping ratio, and $\omega _{n}$ is the natural frequency of the system.

6.2. Environment impedance

An essential element of robot navigation in this research is direct physical contact with its environment. We apply the mechanical impedance principle [Reference Al-Shuka, Leonhardt, Zhu, Song, Ding and Li50] to calculate the contact force interaction based on the impedance coefficient parameter and the tracking position error. Figure 11 illustrates the contact force interaction between the climbing robot and the environment, based on the mechanical impedance principle [Reference Yoshikawa51]. The impedance force function equation of the robot system is then expressed as follows:

Figure 11. Illustration of the environment impedance with the three joint climbing robots.

(34) \begin{align} F_{d}=m_{e}\left(\ddot{x}_{r}-\ddot{x}_{e}\right)+b_{e}\left(\dot{x}_{r}-\dot{x}_{e}\right)+k_{e}\left(x_{r}-x_{e}\right) \end{align}

where m _e , b _e , and k _e represent the impedance parameters (mass, damping, and stiffness) that interacted with the ledge environments; x _e is the environment contact position (ledge), x _r is the reference position (end-effector), and F _d is the impedance interaction force between the robot and the environment. Figure 11 shows the position of the ledge environment is modeled by a mass-spring-damper system. x _e is obtained based on the x _r when contact occurred, which can be written as the following:

(35) \begin{align} \left.\begin{array}{l} x_{e}=\frac{y_{r}-y_{e1}}{h}+x_{e1}\\[3pt] y_{e}=h\left(x_{r}-x_{e1}\right)+y_{e1} \end{array}\right\} \end{align}

where [x _e1 y _e1] ^T and [x _e2 y _e2] ^T are the end tip positions of the ledge, while h is the slope of the ledges and can be written as follows:

(36) \begin{align} h=\frac{\left(y_{e2}-y_{e1}\right)}{\left(x_{e2}-x_{e1}\right)} \end{align}

6.3. Admittance control

Admittance control is a type of position-based impedance control that aims to establish a dynamic relationship between an external force and the robot’s position [Reference Liu and Li52]. The admittance control as illustrated in Fig. 12, allows the robot to make adjustments to changes in its environment.

Figure 12. Illustration of a robot’s trajectory adjustment by admittance control in response to the environment change.

When the robot receives an external disturbance force $\mathit{\tau }_{\mathit{d}}\neq 0$ , the dynamic equation can be expressed as follows:

(37) \begin{align} M_{0}(q)\ddot{q}+C_{0}(q,\dot{q})\dot{q}+G_{0}(q)=\tau +f\left(\tau _{ext},\rho, t\right) \end{align}

The admittance control mechanism relates the force feedback to the system’s position control [Reference Fujiki and Tahara32]. As a result, the system with input F _ext (s) and output x _r (s) can be described as follows:

(38) \begin{align} \left(M_{a}s^{2}+B_{a}s+K_{a}\right)x_{r}\left(s\right)=F_{ext}\left(s\right) \end{align}

where x _r (s) is the revised trajectory of the robot after the system obtains the contact force feedback information with the environment F _ext (s); therefore M _a , B _a , and K _a are the parameters of inertia, damping, and stiffness between the robot and the environment, respectively. In Eq. (38), the dynamic model is expressed in a joint space. So, to apply the admittance control function, it must be expressed in Cartesian space, like in Eq. (39). So, it can be written as follows:

(39) \begin{align} M\left(x\right)\ddot{x}+C\left(x,\dot{x}\right)\dot{x}+G\left(x\right)=F-F_{ext} \end{align}

To obtain a position revision based on contact force feedback information, we need to design the controller input $\mathit{F}$ using equations (39) and (40), which are expressed as follows:

(40) \begin{align} F=C\left(x,\dot{x}\right)\dot{x}+G\left(x\right)+M\left(x\right)\left\{\ddot{x}_{d}-M_{a}^{-1}\left[B_{a}\left(\dot{x}-\dot{x}_{d}\right)+K_{a}\left(x-x_{d}\right)+F_{ext}\right]\right\}\,\, \end{align}

By applying Eq. (41), the admittance control approach can be achieved, allowing the robot to navigate through a direction change of graspable ledge in a safe and smooth transition.

7.1. Searching method

The effectiveness of the searching method inspired by search and maritime rescue has been proven in cases where information is provided only in the search zone [Reference Otote, Li, Ai, Gao, Xu, Chen and G.33]. We adopt searching methods such as sector search, creeping line, A* algorithm, and expanded square. These searching methods are proposed to be modeled in the robot’s workspace, as seen in Fig. 13.

Figure 13. The grasping position search method for a transverse climbing robot: (a) sector search; (b); creeping line; (c) A* algorithm; (d) expanded square.

The proposed methods have been studied for several reasons, such as being simple in the searching region confined to the robot’s workspace [Reference Jiang, Yao, Huang, Yu, Wang and Bi34], having a low computation cost [Reference Majeed and Hwang53], and being acceptable for the specific position of the graspable ledge. Since $d$ is the minimum distance of both hands and $\mathit{\alpha }$ is the span of the searching path line, to preserve the robot’s chance of finding a ledge, the $\mathit{\alpha }$ can be expressed as:

(41) \begin{align} 2\alpha \leq d \end{align}

In this simulation, we systematically assess the performance of each grasping position search method. The assessment criteria are based on the robot’s hand-exploration distance [Reference Niroui, Zhang, Kashino and Nejat54], the total energy consumption [Reference Mei, Lu, Hu and Lee55], and the time duration necessary for executing the grasping position search operation. The total energy consumption of the transverse climbing robot is expressed as follows [Reference Lin and Lee3].

(42) \begin{align} E=\int \left| \tau \right| \,d\theta \end{align}

where $\mathit{E}$ is the total energy consumption, which depends on the joint torque and the angular angle displacement of the robot arm.

7.2. Obstacle model

Figure 14 illustrates an obstacle model that closely represents the actual environment conditions found on a building’s exterior. For the challenge, this obstacle model has a complex arrangement of graspable ledge features. In this testing scenario, the robot begins at a designated start point, after which the motion planning and searching methods are activated to explore and navigate toward the endpoint. The ledge coordinate written in the obstacle model is used by the environment impedance function to calculate the contact force. It is important to note that the robot does not have information about the obstacle path during exploration. Instead, using its own navigation strategies, the robot can identify the environment information at each step of its journey.

Figure 14. Obstacle model (environment map), including position coordinates for environment impedance function.

This scenario in Figure 14 is designed to evaluate the robot’s capability to travel by performing transverse climbing, slope change transitioning, and grasping position search procedures with a variety of obstacles. While the robot has traveled through every position on this obstacle, it is important for the robot to estimate the robot’s cost of transport (COT) [Reference Wongwirat and Anuntachai56]. Each of the robot’s searching strategies will result in unique COT results. While COT is defined as the cost of energy consumption necessary per given unit of weight and unit of moving distance [Reference Nishii57], it is expressed as follows [Reference Lin and Lee3].

(43) \begin{align} COT=\frac{E}{mgd} \end{align}

where m is the robot mass, g is the gravity acceleration, and d is the robot work-done distance.

7.3. Investigation of the momentum observer

Figure 15(a) shows the two observers deployed at the robot joint to detect changes in external torque disturbances. In this scenario, the robot does not generate any dynamic force (no internal force disturbance), but an external torque disturbance is applied with a sinusoidal wave reference. The comparison in Fig. 15(b) shows that SOMO is faster to estimate the actual disturbance response than FOMO. Furthermore, Fig. 15(c) shows that SOMO achieves higher estimation accuracy.

Figure 15. The performance of first-order momentum observer and second-order momentum observer with sinusoidal wave reference; (a) estimation of external torque; (b) details of the graph; (c) external torque estimation error.

When using the force feedback approach, dynamic uncertainty is one of the internal disturbances that cause the robot to misdetect contact with the environment during navigation. Since the dynamic uncertainty value cannot be predicted or measured [Reference Song, Yi, Zhao and Li43], without loss of generality, we assume the dynamic uncertainty deviation is 20% of the modeled dynamic value. Figure 16 demonstrates the robot’s dynamic simulation results, where each joint torque observer is integrated using TVT. In this scenario, the robot generates dynamic forces and uncertainty. The sudden external disturbances simultaneously apply to its dynamic system from 2 s to 2.5 s and 6.5 s to 7 s, indicating collision forces. We set the phase shift coefficient and TVT’s upper and lower bounds at 1 Nm and −1 Nm, respectively. These findings indicate that when no external disturbance is applied, the robot can tolerate the influence of the uncertainty of the dynamic model. Therefore, the system does not detect it as an external disturbance.

Figure 16. The performance of first-order momentum observer and second-order momentum observer with step function reference; (a) estimation of external torque integrated with TVT; (b) details of the graph.

When the estimated external torque applied exceeds the TVT’s upper or lower tolerance bounds, the robot detects an external disturbance. In contrast, Fig. 16(b) shows that SOMO detects sudden external torque with a faster response time and faster convergence than FOMO. Another feature noted in Fig. 17 is that SOMO’s estimation error for external joint torque is smaller. Figures 15 and 16 demonstrate that the performance of SOMO is consistently outstanding. A faster and more accurate response for disturbance estimation may increase the robot’s safety during navigation [Reference Li, Li, Zhu, Xu and Mu28]. Therefore, based on this evaluation, SOMO is an appropriate alternative to be implemented for the robot as a detector and estimator of the contact force with the environment. However, SOMO is an extended state of FOMO, which requires more computational power.

Figure 17. The performance of first-order momentum observer and second-order momentum observer with step function reference; (a) external torque estimation error; (b) details of the graph.

7.4. Simulation in the model obstacle

In this simulation, motion planning and grasping position search strategies are assessed by allowing a transverse climbing robot to perform exploration through an obstacle model. In a preliminary discussion, we analyze the robot’s performance using the sector search approach. Figure 18(a) shows the estimated external joint torque using SOMO, indicating that the robot completed its exploration of the obstacle model in 53.39 s. These spikes indicate a disturbance received by the robot, which should be integrated with the TVT to establish the categorization of the external or internal disturbance. Figure 18(b) shows that the estimated error is obtained based on the difference between the external joint torque estimation and the actual disturbance force. When the robot is in a condition without disturbance, the error is relatively low (nearly 0 Nm), but when the robot gets disturbed, the system suffers a slight inaccuracy when estimating the disturbance force. This suggests that SOMO is operating without difficulties. The simulation video demo can be obtained by visiting this web link: https://youtu.be/9jyP6XNLWps (Accessed on January 12, 2025).

Figure 18. The performance of second-order momentum observer using a sector search method for exploration. (a) external torque estimation; (b) the estimation errors.

A technique for detecting an external disturbance or contact force between the robot and the environment is developed by integrating the SOMO and TVT, as presented in Fig. 19. When the joint torque estimation exceeds the TVT tolerance threshold, the system will detect it as an external disturbance. The results show that the robot has identified a number of contact forces during the obstacle model exploration, each of which is designated by the symbols “a” to “m,” and its explanation of denotation is provided in Table III.

Table III. Identification of recognized external disturbances.

Figure 19. TVT integrated with second-order momentum observer external joint torque estimation and admittance switch.

Figure 20 illustrates the movement and locomotion trace of a transverse climbing robot during exploration through the obstacle model. The robot moved forward from the beginning point to the end destination point with a work-done distance of 3.451 m by navigating autonomously without detecting the current condition of the environment in advance. According to Table III, the robot has completed the environment transition through a six-time grasping position search and a one-time ledge slope change. A proposed motion planning and searching strategy has worked properly according to its desired function. As proven during the searching procedure, the robot detects two contact forces: firstly, to determine the ideal position for executing the searching procedure, and secondly, when it finds the unknown graspable position during the searching procedure.

Following Fig. 20, the robot completed a search procedure across three different ledge gap characteristics. The first characteristic is the upper-ledge gap position, as demonstrated by the 1^st and 2^nd searches. If it’s linked to the actual environment, the robot can climb upward to a horizontal ledge where the position is higher than the robot. The second one is the short-ledge feature gap position characteristics, as demonstrated by the 3^rd and 4^th searches. The last one is the lower-ledge gap position characteristics, as demonstrated in the 5^th and 6^th searches. Where the robot may climb downwards to the ledge where its position is lower than the robot. In addition to executing these searching procedures, the robot is capable of making transitions across a slope change on the connected ledge. For example, a ledge is designed like a roof.

Immediately after the robot detects the contact force with the environment, an automatic admittance switch will respond spontaneously to switch the trajectory reference to the admittance control. Admittance control computes a revised trajectory due to the ledge direction shift based on the estimated contact force given by the observer, resulting in a new motion planning reference. In Fig. 21, it can be noted that when the system detects a contact force (for example, when the grasping ledge was recognized during the searching procedure), the admittance control revises the robot joint angle to adjust it to the change in environment. As a result, the spike in robot joint torque caused by contact spontaneously declines within the TVT tolerance limit. Returning to Fig. 19, it is suggested that an automated admittance system has an essential contribution in assisting in minimizing the amount of collision force acquired by the robot, which may reduce the risk of serious damage to the robot. Based on the algorithm of motion planning and searching strategy, an admittance switch will revert to a disabled state once the admittance adjustment action has been completed.

Figure 20. A transverse movement trace illustrations of a transverse climbing robot’s, including work-done distance, slope change, and grasping position search. It can be demonstrated in the simulation video by accessing this web link: https://youtu.be/9jyP6XNLWps (Accessed on January 12, 2025).

Figure 21. Robot joint control results.

7.5. Performance of grasping position search method

The present study assesses four different methods of grasping position search. Figure 22 shows the transverse climbing robot’s hand-exploration and hand-returning trace using a variety of searching methods. All of the methods have successfully made the robot travel through the obstacle model for the identical total work-done distance of 3.45 m, but each of them has various performance characteristics. Figure 23(a) illustrates the performance of the exploration time relative to the total work-done distance, where the sector search method completes the task in the fastest time, which is 53.39 s. All of the other methods have significant delays in the 2^nd and 3^rd searches (short-ledge feature positions), whereas the sector search method only has slight delays in the 5th and 6th searches (lower-ledge positions).

Figure 22. An exploration trace of a transverse climbing robot by using its grasping position search method; (a) sector search; (b) creeping line; (c) A* algorithm; (d) expanded square.

Figure 23. Performance of grasping position search method; (a) exploration time; (b) hand-exploring distance; (c) energy consumption; (d) cost of transport.

Figure 23(b) illustrates the hand-exploration distance relative to the work-done distance, revealing that the sector search method requires the shortest exploration path distance, which is 10.13 m. All of the other methods require a considerably increased exploration path distance for the 2^nd and 3^rd searches, which causes the exploring time to get longer. If we observe the exploration path of the sector search method in Fig. 22(a), its searching path is not simply focused on one specified location that must be finished early on. As a result, the sector search method is getting faster to reach the workspace’s center region, which makes it easier to execute searching procedures in the short-ledge feature position.

Figure 23(d) illustrates that the A* algorithm consumes the least amount of total energy. Although the A* algorithm also has difficulties in the 2^nd and 3^rd searches, it just spends the least amount of energy compared to the creeping line and extended square. This is because the searching path of the A* algorithm focuses on tracking the workspace boundary line. Therefore, after reaching the upper boundary location, it will track to the lower boundary, or vice versa. This is distinct from the creeping line method, which has spent the largest amount of energy in the 2^nd and 3^rd searches due to its searching path line focusing on tracing the higher and lower parts of the workspace practically at the same time by following the horizontal centerline of the workspace. However, because of its searching path characteristics, the creeping line method is the most successful for finding the lower ledge position. Figure 23(c) illustrates the cost of transport (COT), with the A* algorithm being the least costly compared to the other methods. The cost of transport is related to the amount of energy consumed per unit of work-done distance.

The next assessment is to identify the most effective grasping position search method for each gap characteristic between graspable ledges. Table IV(a) presents the searching performance on the upper-ledge position (at the 1^st and 4^th searches), where the sector search method is the most effective in terms of searching time, hand-exploration distance, and energy consumption. Next is Table IV(b), which presents its searching performance on the short-edge features position (at the 2^nd and 3^rd searches), where the sector search method still has outstanding performance in the same criteria. Finally, Table IV(c) presents the search performance on the lower-ledge position (at the 5^th and 6^th searches), where the creeping line method is the most effective in terms of searching time, hand-exploration distance, and energy consumption. Meanwhile, sector search is ranked third.

Table IV. Performance of the grasping position search method based on characteristics of the obstacle; (a) upper-ledge position; (b) short-ledge features; (c) lower-ledge position.

The four types of grasping position search methods that have been assessed in this study have varying performances in each aspect of the gaps between grasping ledges. So, in practice, it is difficult to integrate more than one grasping position search method during a single environment exploration. Because, in fact, the robot does not have any information about what’s occurring in front of it when exploring the environment. A future study focusing on developing an optimization of its searching method could be a solution to this challenge. However, based on the discussion of this present study, we make a decision on how to determine the compatibility of the grasping position search method by considering the following desires: First, for faster exploration time, we recommend using the sector search method. This can be examined in Fig. 21(a), where the rise in exploration time is substantially more stable during the searching and travel climbing procedures. The sector search method is also recommended for a shorter exploration path length, as can be noticed in Fig. 21(b). Furthermore, to reduce energy consumption and COT, we recommend using the A* algorithm. This can be observed in Fig. 21(d), where the increase in consumption of energy seems remarkably consistent. In addition, if assessed on the searching performance of all gap characteristics between the grasping ledges, which is presented in Table IV(a), (b), and (c), the A* algorithm is consistently in the second rank in terms of energy consumption.

7.6. Comparative review of visual-based and contact force-based navigation

This Section presents a comparative review of the four-link transverse ledge climbing robot, comparing the contact force-based navigation method developed in this research with the visual-based navigation method introduced by Lin et al. [Reference Lin and Hu2]. This comparison highlights how the contact-force-based approach addresses the limitations of the visual-based method. The results of this comparison analysis are summarized in Table V.

Table V. Comparison of visual-based and force-based navigation methods for a four-link ledge transverse climbing robot.

The climbing robot developed in this research can perform exploratory movements across its workspace by dynamically and repeatedly attempting contact with the environment, resulting in a wider sensing range for a variety of obstacles. If the robot interacts with a rigid environment, it can acquire an optimal contact impedance force for admittance control references to accurately calculate the robot’s position as the environment changes. So, the environment’s surface reflection does not affect the contact impedance force. By utilizing momentum observers, the robot’s space requirements for the installation of additional sensor devices that may limit the robot’s movement can be eliminated, allowing for more agile movement and enhanced adaptation to complex environments.